From 276109511819c53e58c88858cb283a5a6a031746 Mon Sep 17 00:00:00 2001 From: Kushala Rani Talakad Manjunath Date: Sun, 29 Mar 2026 20:25:40 -0400 Subject: [PATCH] Fix errors to files. Fix minor errors to LaTeX file and save the changes to corresponding pdf file. --- VisualizingAMatrix.pdf | Bin 1206150 -> 1218153 bytes VisualizingAMatrix.tex | 117 +++++++++++++++++++++-------------------- 2 files changed, 59 insertions(+), 58 deletions(-) diff --git a/VisualizingAMatrix.pdf b/VisualizingAMatrix.pdf index 4487007af01aaa1a0c450fa3c264918510dc67e2..89c2681afcf1415876ece36824cf25752b59358c 100644 GIT binary patch delta 88249 zcmZs>LzE^=)3sZ+ZQHhOn_afeyKLKLmu=g&Z5v&6p7(sye~>G45-VqM?bxxy=AqYNoVS6kz4>Tvo zI5h#E#2N<3XjUoSoNWdri$to4>E_Z%gX1&1K3Bib-!Iu$eS*JlPw8_Uy16#LIMM3> zryc;c%J1pQ6Tr^D+>G(=t z(AA6`j8?99heG;L5Yj1WThOhHb4DWkCjfw-xtEPNo_;Zz#ryhtrew7=KzE+5{nlPI z92cJF6P7lG*h)CH`O~^Vv*)NeG17p?^ObnTt26oX*Ur0(GuQ66eF%Y{6%u5<^*}+B z#bagOz43vPAz-5Y3R1gWxwkkF2Pk?tkO9U)7t9H#Db9K;qcK?9>NdDB>!}+T2pnoWsnl|OI zA7w~OFvh1L=z4`v$97lNg)1evDsZ6P+hVgAI9EdsgT0U3N;FLs52;4Eh$InXemk~x 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z`q5){R=jz7@;5BMs~auYvHq$5JHdx`)oQAr`Zc-M&sK1$Ot3f(x!g z2G{zt*og+t>7TTzdD1^#F%V)GKBF6PtyX7wfy;E$&m#6A_m;B#`F;tcYHSj;>tl&(ct5NJwb_4G4#E>|H*u&0SuBGq! z0LL9aakJNEzAjNnV1bOUFYOc(+%eQErUc|>HrKn#k_Bc1x|L$Bx3$&!xV$yjF5?_ti; zBP5#k#8@RTsMuu=dqjlv&U1*Va?)I%7)Z)o3uofxq!~ycl6Fnk_{z7Jl3K5aKl2B) zeo)d5a#x^ksLa_XyNJ*q*N*Ia;JWGU7r(CjVyXDsA*> test\_cond\_thm(100) \\ @@ -991,7 +990,7 @@ \subsection{Application: Sensitivity of linear solve} } As can be seen, the inferred condition number is bounded by $\kappa = 100$ above. That is, although at least one trial got above 60, the ratio clearly never exceeded 100. This demonstrates that the inequality -\cref{condnothm} is obeyed, and shows +\eqref{condnothm} is obeyed, and shows how the condition number characterizes the sensitivity of the linear solve operation to the effects of input error or perturbations. @@ -1017,7 +1016,7 @@ \section{The quadratic form visualization} 2D plane, then the resulting $f(u)$ is a surface floating above the plane. -Although one my create a quadratic form using any arbitrary matrix, for +Although one may create a quadratic form using any arbitrary matrix, for what follows we will restrict ourselves to symmetric matrices $A$ with all real elements. The reasons for this are: \begin{itemize} @@ -1073,8 +1072,8 @@ \subsection{Eigenvalues and eigenvectors} \includegraphics[width=1.0\columnwidth]{QuadraticFormFixedSurfaces.png} \caption{From left to right: 1. Upward opening paraboloid: the matrix is positive definite, both eigenvalues are positive. 2. Downward opening paraboloid: the matrix is - negative definite, both eigenvalues are negative. 3. Saddle: The matrix is indeterminate, - one eigenvalue is positive, one is negative. 4. Cylidrical paraboloid: The matrix is + negative definite, both eigenvalues are negative. 3. Saddle: The matrix is indefinite, + one eigenvalue is positive, one is negative. 4. Cylindrical paraboloid: The matrix is singular, one eigenvalue is positive, the other is zero.} \label{fig:QuadraticFormFixedSurfaces} \end{figure} @@ -1085,7 +1084,7 @@ \subsection{Eigenvalues and eigenvectors} \label{eq:evd} A = Q \Lambda Q^T \end{equation} -where $Q$ is an orthonormal matrix, and $\Lambda$ is a diagonal matrix whose diagonal elements +where $Q$ is an orthogonal matrix, and $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues of matrix $A$. I will assume you have some basic familiarity with eigenvalues and eigenvectors; the subject is well treated in the standard linear algebra textbooks. One important thing to note is that by restricting ourselves to real, symmetric @@ -1104,8 +1103,8 @@ \subsection{Eigenvalues and eigenvectors} eigenvalue corresponds to a downward-facing parabola. If an eigenvalue is zero, it corresponds to one direction of the surface which faces neither up nor down -- this is the degenerate case shown in \cref{fig:QuadraticFormFixedSurfaces} - \item The two directions of principle curvature are given by the eigenvectors of $A$. The - term "principle curvature" means the minimum and maximum values of curvature along all + \item The two directions of principal curvature are given by the eigenvectors of $A$. The + term "principal curvature" means the minimum and maximum values of curvature along all lines passing through the origin. \item A positive definite matrix is one whose eigenvalues are all positive. Such matrices show up frequently in real-world problems. For a 2x2 matrix the corresponding surface is a paraboloid opening @@ -1123,7 +1122,7 @@ \subsection{Eigenvalues and eigenvectors} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %XXXX Need drawing showing vertical plates cutting paraboloid giving parabolas -%XXXX Need drawing showing how eigenvectors align along directions of principle curvature. +%XXXX Need drawing showing how eigenvectors align along directions of principal curvature. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -1136,17 +1135,17 @@ \subsection{Relation between ellipse and quadratic form visualizations} lines of intersection between the paraboloid and the slicing planes are called "levelsets" in optimization speak, or the "isolines" in computer graphics speak. Importantly for us, the lines are manifestly ellipses -- a fact that should -immediately remind us of the ellipse visualization presented in \cref{sec:unit_ball}. +immediately remind us of the ellipse visualization presented in \eqref{sec:unit_ball}. \begin{figure}[thb] \centering \includegraphics[width=0.7\columnwidth]{LevelsetsOfParabola.png} - \caption{The levelsets (isolines) of the sliced paraboliod are + \caption{The levelsets (isolines) of the sliced paraboloid are ellipses, shown here as solid green and blue lines corresponding to different slicing planes.} \label{fig:LevelsetsOfParabola} \end{figure} The natural question is, is there a relationship between these ellipses and -the ellipses generated by matrix-vector multiplication studied in \cref{sec:unit_ball}? +the ellipses generated by matrix-vector multiplication studied in \eqref{sec:unit_ball}? The answer is yes, the isoline ellipses depicted in \cref{fig:LevelsetsOfParabola} and the ellipses depicted in \cref{fig:CircleTransformedToEllipse} are indeed related. @@ -1163,16 +1162,18 @@ \subsection{Relation between ellipse and quadratic form visualizations} Next imagine that $A$ may be decomposed into a product $A = B^T B$. Since $A$ is positive definite we know this is the case -- the decomposition is essentially the Cholesky decomposition \cite{Higham2020_Chol}. -Using this, \cref{eq:quad_form} becomes +Using this, \eqref{eq:quad_form} becomes \begin{equation} -1 = u^T B^T B u \\ -= (u^T B^T) (B u) \\ -= e^T e +\begin{aligned} +1 = u^T B^T B u += (u^T B^T) (B u) += e^T e +\end{aligned} \end{equation} where $e = B u$ is a vector. This equation says that the 2-norm of $e$ is one. Therefore, the set of all vectors $e$ are unit vectors, and their tips lie on the the unit circle. -This was the starting point of the ellipse visualization back in section \cref{sec:unit_ball}! That -means the ellipses defined by \cref{eq:quad_form} are created by the matrix-vector +This was the starting point of the ellipse visualization back in section \eqref{sec:unit_ball}! That +means the ellipses defined by \eqref{eq:quad_form} are created by the matrix-vector multiplication $$ u = B^{-1} e @@ -1181,15 +1182,17 @@ \subsection{Relation between ellipse and quadratic form visualizations} So the the isoline ellipses and the ellipses generated from the unit circle under matrix-vector multiplication are indeed related. But can we find a straightforward -forumula connecting them? Consider the image of the unit circle under $A$, +formula connecting them? Consider the image of the unit circle under $A$, $$ w = A e $$ where $e$ is the unit circle and $w$ is its image as an ellipse. Knowing that, and using the definition of $B$ we have $$ -w = B^T B e \\ +\begin{aligned} +w = B^T B e = B^T B B u +\end{aligned} $$ This says that $w$ is the image of $u$ under matrix-vector multiplication. In particular, the first product $B u$ returns a unit circle, and then $B^T B$ transforms it into @@ -1213,7 +1216,7 @@ \subsection{Relation between ellipse and quadratic form visualizations} \end{figure} \subsection{Condition number and the quadratic form visualization} -Recall using the ellipse visualization in \cref{conditionnumber} as +Recall using the ellipse visualization in \eqref{conditionnumber} as a proxy of the condition number. Can we also use the quadratic form visualization as a way to understand the condition number of $A$? As usual, when I ask a rhetorical question in this article the answer is "yes". @@ -1400,9 +1403,7 @@ \subsection{Condition number theorem} \bibitem{Higham2020_BkwdErr} Nick Higham, "What Is Backward Error?", \url{https://nhigham.com/2020/03/25/what-is-backward-error/} -\bibitem{Trefethen97} For a full discussion, see L. Trefethen, and D. Bau, -"Numerical Linear Algebra", -SIAM, Philadelphia, (1997) +\bibitem{Trefethen97} L. Trefethen, and D. Bau, "Numerical Linear Algebra", SIAM, Philadelphia, (1997) \bibitem{Higham2020_Chol} Nick Higham, "What Is a Cholesky Factorization?", \url{https://nhigham.com/2020/08/11/what-is-a-cholesky-factorization/}

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